Here’s the analysis and solution to the activity:

Part A:

1. Four coins are tossed (Random variable X represents the number of tails):

   • Possible outcomes for 4 coins are:
     {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, TTHH, HTTH, THHT, THTH, TTTH, HTTT, THTT, TTHT, TTTT}.
     • Number of tails (X) in each outcome ranges from 0 to 4.
   • Values of X: {0, 1, 2, 3, 4}

2. Three balls are drawn without replacement (Random variable Z represents the number of blue balls):

   • The total composition of balls is:
     5 red, 6 yellow, 6 blue (Total = 17 balls).
     • Since three balls are drawn, Z (number of blue balls) can take values:
       {0, 1, 2, 3}.
       The values depend on the combination of draws (e.g., no blue balls drawn, exactly 1 blue ball, etc.).

Part B: Write all possible values of each random variable:

1. X: Number of odd-number outcomes in a roll of two dice.

   • Possible outcomes for one die: {1, 2, 3, 4, 5, 6}.
     Odd outcomes: {1, 3, 5}.
     • For two dice, the number of odd outcomes can range from 0 (no dice showing odd) to 2 (both dice showing odd).
       Values of X: {0, 1, 2}.

2. Z: Scores of a student in a 15-item test.

   • Possible scores range from 0 (no correct answers) to 15 (all correct).
     Values of Z: {0, 1, 2, ..., 15}.

3. Y: Height (in ft) of a basketball player not exceeding 7 ft.

   • Heights can vary continuously between a lower bound and 7 ft.
     Values of Y: Any real number in the interval [0, 7] (continuous variable).

4. W: Product of two numbers taken from two boxes containing numbers 1 to 6.

   • Possible products are the results of multiplying any pair of numbers from {1, 2, 3, 4, 5, 6}.
     Products: {1, 2, 3, 4, 5, 6, 4, 6, 8, 9, 10, 12, ..., 36} (all unique combinations).
     Values of W: {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, ..., 36}.

Part C: Classify each random variable as discrete or continuous:

1. Score of a student in a quiz:

   • Discrete (Scores are countable integers).

2. How long students ate breakfast:

   • Continuous (Time is measured, not counted).

Follow-up Questions:

1. How would the probabilities be calculated for X in part A1?
2. Can Z in part A2 change if the composition of balls changes?
3. What is the maximum possible value for W in part B4?
4. How do you classify other variables not explicitly mentioned as discrete or continuous?
5. What are the practical implications of distinguishing discrete and continuous variables?

Tip:

When analyzing random variables, always check the nature of their outcomes: countable (discrete) or measurable on a continuous scale (continuous).